Homogeneous Maps and Divisorial Ideals

Given anR-moduleV, anR-homogeneous mapf:V → Vis a function that is linear on all cyclic submodules ofV. For each positive integern, denote by nthe integral domains with the property that for each homogeneous mapf:Rn → Rnand for each cyclic submoduleC Rn, there exists a linear map σ:Rn → Rnsuch thatFC = σC. Necessary and sufficient conditions forRto be in n, in terms of divisorial ideals, are derived.